Transport for London figures show there were 34 incidents involving the Barclays Cycle Hire bikes between 30 July and 30 September.
There were also five incidents related to the Boris bike scheme — including one where a woman was knocked off her cycle by one of the trailers used to transport the bikes between docking stations.
One cyclist suffered a head injury when he was hit by a bus and a seven-year-old narrowly escaped injury when a docking station fell on the child after it was hit by a car.
In the first recorded accident, five days into the scheme, a male hire bike user was squashed against a kerb by a lorry.
According to the Royal Society for the Prevention of Accidents, 3,674 cyclists, or 10 per day, were killed or injured in London last year.